Determination of the characteristic variety V(I_k) for V^k(sl3) at level k = -3 + 2/(2m+1)

Determine the Harish–Chandra characteristic variety V(I_k), as defined via the image of the maximal ideal I_k of V^k(sl_3) in the Zhu’s algebra A(V^k(sl_3)) ≅ U(sl_3), at levels k = -3 + 2/(2m+1) with m > 0, by proving that V(I_k) equals the set {tΛ_1 − (2i/(2m+1))Λ_3, tΛ_2 − (2i/(2m+1))Λ_3, tΛ_1 − ((t + 2i + 1)/(2m+1))Λ_3 | t ∈ C, i = 0,1,...,2m}.

Background

In Section 2, the paper recalls the notion of the characteristic variety V(I) for a two-sided ideal I ⊂ U(g) defined via the Harish–Chandra projection. For the maximal ideal I_k of the affine vertex algebra V^k(sl_3), its image in A(V^k(sl_3)) ≅ U(sl_3) determines V(I_k).

The conjecture proposes an explicit description of V(I_k) as a finite union of affine lines in weight space parameterized by t ∈ C and indexed by i = 0,...,2m, tightly connected to the representation theory of L_k(sl_3) and to the proposed classification of A(L_k(sl_3))-modules in Conjecture 1.5.